In each of the following questions , three statements of numbers following same rules are given . Find the rule and accordingly find the value of the number ?
If `213 = 419, 322 = 924, 415 = 16125`, then 215 = ?

Options

A. 425
B. 1625
C. 4125
D. 2541

Explanation

To solve the problem, we need to identify the pattern in the given statements and then apply that pattern to find the value of `215`.

### Step-by-Step Solution:

1. **Analyze the first statement: `213 = 419`**
- Break down the numbers:
- The first digit is `2`, the second digit is `1`, and the third digit is `3`.
- The result `419` can be seen as:
- The square of the first digit: \(2^2 = 4\)
- The middle digit remains the same: \(1\)
- The square of the last digit: \(3^2 = 9\)
- So, we can write: \(2^2, 1, 3^2 \Rightarrow 4, 1, 9\)

2. **Analyze the second statement: `322 = 924`**
- Break down the numbers:
- The first digit is `3`, the second digit is `2`, and the third digit is `2`.
- The result `924` can be seen as:
- The square of the first digit: \(3^2 = 9\)
- The middle digit remains the same: \(2\)
- The square of the last digit: \(2^2 = 4\)
- So, we can write: \(3^2, 2, 2^2 \Rightarrow 9, 2, 4\)

3. **Analyze the third statement: `415 = 16125`**
- Break down the numbers:
- The first digit is `4`, the second digit is `1`, and the third digit is `5`.
- The result `16125` can be seen as:
- The square of the first digit: \(4^2 = 16\)
- The middle digit remains the same: \(1\)
- The square of the last digit: \(5^2 = 25\)
- So, we can write: \(4^2, 1, 5^2 \Rightarrow 16, 1, 25\)

4. **Identify the rule:**
- The pattern observed is:
- For a number represented as `abc`, the result is formed by:
- \(a^2\) (square of the first digit)
- The middle digit remains the same
- \(c^2\) (square of the last digit)
- Therefore, the format is: \(a^2, b, c^2\)

5. **Apply the rule to `215`:**
- Break down `215`:
- The first digit is `2`, the second digit is `1`, and the third digit is `5`.
- Now apply the rule:
- First digit: \(2^2 = 4\)
- Middle digit: \(1\)
- Last digit: \(5^2 = 25\)
- Combine these results: \(4, 1, 25\)

6. **Final result:**
- Therefore, `215` equals `4125`.

### Final Answer:
**215 = 4125**